Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3565 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7,10,11)(8,14,15,12)(9,17,18,13)(16,19,21,20)(22,23)(24,29)(25,30)(26,34)(27,28)(31,35)(32,33)(36,37);(7,11)(8,15)(13,17)(16,21)(23,28)(25,29)(31,34)(33,36);(6,8,10,15)(7,12,11,14)(9,16,18,21)(13,19,17,20)(22,24,27,30)(23,29,28,25)(26,32,35,37)(31,36,34,33);(6,7)(8,12)(9,13)(10,11)(14,15)(16,19)(17,18)(20,21)(22,25,27,29)(23,30,28,24)(26,33,35,36)(31,37,34,32);(6,9,10,18)(7,13,11,17)(8,16,15,21)(12,19,14,20)(22,26,27,35)(23,31,28,34)(24,32,30,37)(25,33,29,36)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21432);